Derivative of this application?
$begingroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
$endgroup$
add a comment |
$begingroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
$endgroup$
add a comment |
$begingroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
$endgroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
sequences-and-series multivariable-calculus derivatives convergence exponential-function
asked Jan 18 at 1:50
MamanMaman
1,196722
1,196722
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077745%2fderivative-of-this-application%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077745%2fderivative-of-this-application%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown